Notation The identity map on an arbitrary space will be denoted by Id . Whenever X is a set, we write 1X (x) = 1 if x X, 1X (x) = 0 otherwise. The complement of a set A will be denoted by Ac. Throughout the text, whenever we write Rn the dimension n is an ar- bitrary integer n 1. Whenever A is a Lebesgue-measurable subset of Rn, its n-dimensional Lebesgue measure will be denoted by |A|. This should not be confused with the Euclidean norm of a vector x Rn, which will also be denoted by |x|. Whenever x, y Rn we write x · y = x, y = ∑n i=1 xiyi. Given some abstract measure space X, we shall denote by P (X) the set of all probability measures on X, and by M(X) the set of all finite signed measures on X (i.e. precisely the vector space generated by P (X)). The space M(X) is equipped with the norm of total variation, μ TV = inf{μ+[X] + μ−[X]}, where the infimum is taken over all nonnegative mea- sures μ+, μ− such that μ = μ+ μ−. The infimum is obtained when μ+ and μ− are singular to each other, in which case μ = μ+ μ− is said to be the Hahn decomposition of μ. Of course, if ν is a nonnegative measure and f a measurable map, then f L1(dν) = TV . From Chapter 1 on, we shall only work in topological spaces, equipped with their Borel σ-algebra so P (X) will be the set of Borel probability measures. We shall sometimes write w∗−P (X) for P (X) equipped with the weak topology. The Dirac mass at a point x will be denoted by δx: δx[A] = 1 if x A, 0 otherwise. If a particular measure μ on X is singled out, for p [1, ∞) we shall denote by Lp(X) or Lp(dμ) the Lebesgue space of order p for the reference xiii
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